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Asynchronous Behavior of Doublequiescent Elementary Cellular Automata
"... Abstract. In this paper we propose a probabilistic analysis of the relaxation time of elementary finite cellular automata (i.e., {0, 1} states, radius 1 and unidimensional) for which both states are quiescent (i.e., (0, 0, 0) ↦ → 0 and (1, 1, 1) ↦ → 1), under αasynchronous dynamics (i.e., each cell ..."
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Cited by 7 (1 self)
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Abstract. In this paper we propose a probabilistic analysis of the relaxation time of elementary finite cellular automata (i.e., {0, 1} states, radius 1 and unidimensional) for which both states are quiescent (i.e., (0, 0, 0) ↦ → 0 and (1, 1, 1) ↦ → 1), under αasynchronous dynamics (i.e., each cell is updated at each time step independently with probability 0 < α � 1). This work generalizes previous work in [1], in the sense that we study here a continuous range of asynchronism that goes from full asynchronism to full synchronism. We characterize formally the sensitivity to asynchronism of the relaxation times for 52 of the 64 considered automata. Our work relies on the design of probabilistic tools that enable to predict the global behaviour by counting local configuration patterns. These tools may be of independent interest since they provide a convenient framework to deal exhaustively with the tedious case analysis inherent to this kind of study. The remaining 12 automata (only 5 after symmetries) appear to exhibit interesting complex phenomena (such as polynomial/exponential/infinite phase transitions). 1
All kbounded policies are equivalent for selfstabilization
 IN "EIGHTH INTERNATIONAL SYMPOSIUM ON STABILIZATION, SAFETY, AND SECURITY OF DISTRIBUTED SYSTEMS (SSS
, 2006
"... We reduce the problem of proving the convergence of a randomized selfstabilizing algorithm under kbounded policies to the convergence of the same algorithm under a specific policy. As a consequence, all kbounded schedules are equivalent: a given algorithm is selfstabilizing under one of them if ..."
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Cited by 3 (2 self)
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We reduce the problem of proving the convergence of a randomized selfstabilizing algorithm under kbounded policies to the convergence of the same algorithm under a specific policy. As a consequence, all kbounded schedules are equivalent: a given algorithm is selfstabilizing under one of them if and only if it is selfstabilizing under any of them.
Playing With Population Protocols
 in "International Workshop on The Complexity of Simple Programs, Irlande Cork", 2008, http://hal.inria.fr/inria00330344/en/. Activity Report INRIA 2008 International PeerReviewed Conference/Proceedings
"... Population protocols have been introduced as a model of sensor networks consisting of very limited mobile agents with no control over their own movement: A collection of anonymous agents, modeled by finite automata, interact in pairs according to some rules. Predicates on the initial configurations ..."
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Cited by 2 (1 self)
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Population protocols have been introduced as a model of sensor networks consisting of very limited mobile agents with no control over their own movement: A collection of anonymous agents, modeled by finite automata, interact in pairs according to some rules. Predicates on the initial configurations that can be computed by such protocols have been characterized under several hypotheses. We discuss here whether and when the rules of interactions between agents can be seen as a game from game theory. We do so by discussing several basic protocols. 1
Computing with Pavlovian Populations ⋆
"... Abstract. Population protocols have been introduced by Angluin et al. as a model of networks consisting of very limited mobile agents that interact in pairs but with no control over their own movement. A collection of anonymous agents, modeled by finite automata, interact pairwise according to some ..."
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Cited by 1 (1 self)
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Abstract. Population protocols have been introduced by Angluin et al. as a model of networks consisting of very limited mobile agents that interact in pairs but with no control over their own movement. A collection of anonymous agents, modeled by finite automata, interact pairwise according to some rules that update their states. Predicates on the initial configurations that can be computed by such protocols have been characterized as semilinear predicates. In an orthogonal way, several distributed systems have been termed in literature as being realizations of games in the sense of game theory. We investigate under which conditions population protocols, or more generally pairwise interaction rules, correspond to games. We show that restricting to asymetric games is not really a restriction: all predicates computable by protocols can actually be computed by protocols corresponding to games, i.e. any semilinear predicate can be computed by a Pavlovian population multiprotocol. 1
Faulttolerant and Selfstabilizing Mobile Robots Gathering  Feasibility Study 
"... Publication interne n 1802 may 2006  20 pages Abstract: Gathering is a fundamental coordination problem in cooperative mobile robotics. In short, given a set of robots with arbitrary initial location and no initial agreement on a global coordinate system, gathering requires that all robots, follow ..."
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Publication interne n 1802 may 2006  20 pages Abstract: Gathering is a fundamental coordination problem in cooperative mobile robotics. In short, given a set of robots with arbitrary initial location and no initial agreement on a global coordinate system, gathering requires that all robots, following their algorithm, reach the exact same but not predetermined location. In this paper, we signi cantly extend the studies of deterministic gathering feasibility under di erent assumptions related to synchrony and faults (crash and Byzantine). Unlike prior work, we consider a larger set of scheduling strategies, such as bounded schedulers, and derive interesting lower bounds on these schedulers. In addition, we extend our study to the feasibility of probabilistic gathering in both faultfree and faultprone environments. To the best of our knowledge our work is the rst to address the gathering from a probabilistic point of view. Keywords: mobile robots, faulttolerance, selfstabilization, selforganization, impossibility results, deterministic gathering, probabilistic gathering
Formal Aspects of Computing Probabilistic Verification of Herman’s SelfStabilisation Algorithm
"... Abstract. Herman’s selfstabilisation algorithm provides a simple randomised solution to the problem of recovering from faults in an Nprocess token ring. However, a precise analysis of the algorithm’s maximum execution time proves to be surprisingly difficult. McIver & Morgan have conjectured that ..."
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Abstract. Herman’s selfstabilisation algorithm provides a simple randomised solution to the problem of recovering from faults in an Nprocess token ring. However, a precise analysis of the algorithm’s maximum execution time proves to be surprisingly difficult. McIver & Morgan have conjectured that the worstcase behaviour results from a ring configuration of 3 evenly spaced tokens, giving an expected time of approximately 0.15N 2. However, the tightest upper bound proved to date is 0.64N 2. We apply probabilistic verification techniques, using the probabilistic model checker PRISM, to analyse the conjecture, showing it to be correct for all sizes of the ring that can be exhaustively analysed. We furthermore demonstrate that the worstcase execution time of the algorithm can be reduced by using a biased coin. Keywords: Selfstabilisation; Herman’s algorithm; Probabilistic model checking 1.
On Stabilization in Herman’s Algorithm
, 1104
"... Abstract. Herman’s algorithm is a synchronous randomized protocol for achieving selfstabilization in a token ring consisting of N processes. The interaction of tokens makes the dynamics of the protocol very difficult to analyze. In this paper we study the expected time to stabilization in terms of ..."
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Abstract. Herman’s algorithm is a synchronous randomized protocol for achieving selfstabilization in a token ring consisting of N processes. The interaction of tokens makes the dynamics of the protocol very difficult to analyze. In this paper we study the expected time to stabilization in terms of the initial configuration. It is straightforward that the algorithm achieves stabilization almost surely from any initial configuration, and it is known that the worstcase expected time to stabilization (with respect to the initial configuration) is Θ(N 2). Our first contribution is to give an upper bound of 0.64N 2 on the expected stabilization time, improving on previous upper bounds and reducing the gap with the best existing lower bound. We also introduce an asynchronous version of the protocol, showing a similar O(N 2) convergence bound in this case. Assuming that errors arise from the corruption of some number k of bits, where k is fixed independently of the size of the ring, we show that the expected time to stabilization is O(N). This reveals a hitherto unknown and highly desirable property of Herman’s algorithm: it recovers quickly from bounded errors. We also show that if the initial configuration arises by resetting each bit independently and uniformly at random, then stabilization is significantly faster than in the worst case. 1
Distributed Computing manuscript No. (will be inserted by the editor) Randomized Selfstabilizing and Space Optimal Leader Election under Arbitrary Scheduler on Rings
"... The date of receipt and acceptance will be inserted by the editor Summary. We present a randomized selfstabilizing leader election protocol and a randomized selfstabilizing token circulation protocol under an arbitrary scheduler on anonymous and unidirectional rings of any size. These protocols ar ..."
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The date of receipt and acceptance will be inserted by the editor Summary. We present a randomized selfstabilizing leader election protocol and a randomized selfstabilizing token circulation protocol under an arbitrary scheduler on anonymous and unidirectional rings of any size. These protocols are space optimal. We also give a formal and complete proof of these protocols. To this end, we develop a complete model for probabilistic selfstabilizing distributed systems which clearly separates the non deterministic behavior of the scheduler from the randomized behavior of the protocol. This framework includes all the necessary tools for proving the selfstabilization of a randomized distributed system: definition of a probabilistic space and definition of the selfstabilization of a randomized protocol. We also propose a new technique of scheduler management through a selfstabilizing protocol composition (crossover composition). Roughly speaking, we force all computations to have a fairness property under any scheduler, even under an unfair one.